Tag Archives: christopher alexander

I was invited to the complex systems laboratory of the Université de Montréal this week to present emergent urbanism to their twenty-member large research group. Click through to SlideShare in order to see the full text of the presentation under the “notes on” tab. The entire text is in French, however I know a significant share of this website’s visitors enjoy French once in a while.

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If someone wants to sponsor me for a translation in English, email me and I’ll upload one very soon. Otherwise my hands are quite full at the moment, it might be a while before I get around to it.

Thanks to Rodolphe Gonzales from the Complex Systems Lab for the invitation. You can read about their work here.

I often wonder if it would be possible to do any kind of serious study into urban morphology without the help of Google Earth. I know it has been indispensable to my studies, perhaps as indispensable as the microscope is to biologists. Google Earth is our macroscope, it allows us to see what is too large to see with the naked eye. But no matter how useful satellite photography is, you cannot truly see depth without aerial photography, and the master of aerial photography is without a doubt French photographer Yann Arthus-Bertrand, famous for enormous coffee-table books filled with photography so rich as to be overwhelming.

Arthus-Bertrand has made the jump to high-definition cinematography and directed a “documentary” (there is really no accurate way to describe this film) called Home, which was released free of charge on the Internet a few weeks ago. You can watch it on YouTube or download it from your favorite BitTorrent source. The film is awe-inspiring. Here are some still images I extracted.

The film is a tour of Earth’s ecologies, starting from elementary life to cities. The most striking images are those of natural cities, particularly one which seems to grow out of the rock as if it were only a feature of it. And who can argue that it isn’t? But that detail seems to escape the narrative.

Yann Arthus-Bertrand’s talent is undeniable, at some point in the film he even makes Manhattan seem small. But the film quickly turns into a vessel of green propaganda (sponsored by the Gucci fashion house) while it could have been a celebration of mankind’s ties to nature. At the climax of its alarmism, the foundation of the green mindset is spelled out with as much emphasis as the narrator can apply. Denouncing the thaw of Siberian permafrost, the narrator recites “if the permafrost melts, the methane released would cause the greenhouse effect to race out of control with consequences no one can predict.” At no point in the film does the alarm in her voice sound so grave. It is not so much climate change that is feared, but the unknown, any change at all.

The obsession with control and prediction is tragically what has caused the most destruction and chaos in our human ecologies. It is control that dictates that homes may not be owned in the world’s sprawling slums, in the name of upholding a failed prediction, city planning. Because slum homes can be summarily demolished the slums remain in squalid poverty, vulnerable to any environmental change, man-made or not. The only true sustainability in a chaotic world is the ability to renew our environments for any change we meet, and control and prediction are an obstacle to this.

Green politics fails not because it relies on facts that are incorrect, but because it relies on facts that are inherently unknowable. We can sound the alarm about the total global population, the fact is we have absolutely no idea what the total global population is. We can at best obtain an estimate, but that estimate is useless for any kind of action. Action in a complex system is local and does not rely on global knowledge, but only on reacting to local conditions. The environment always tells you what to be doing in the moment.

In its obsession with control, the film ends up making recommendations for creating the same kind of technocratic utopia that was promised to us by the modernists. It praises one of the world’s poorest countries for having one of the most intensive state schooling program, evading a causal link between poverty and control of children’s minds. The ultimate solution to climate change proposed is to cover the world’s open land with solar panels, and the seas with wind farms, an act that would be as destructive to the environment as all the other monocultures denounced in the film. (And no one dares ask where those solar panels came from.)

Never is a serious look taken at the process of the natural cities, which to someone trapped in the paradigm of control and prediction would make absolutely no sense, but which Christopher Alexander masterfully demystifies in The Nature of Order. Only through such a revolution can we avoid repeating the chaos of modernism with a green twist.

People trapped in the mindset of prediction cannot think beyond simple physical processes (type I and II of Wolfram’s classification). These processes are always highly unstable and prone to die with any disruption. But life is not a simple process. It is a process that is always expanding, growing exponentially to fill any space it can fit into. Biologists quarantined a volcanic island that appeared into existence in the 1960’s near Iceland. They wanted to see how life colonized it. This process has taken place at astonishing speed, and today the island teems with life and has a rich cover of top soil, bewildering the biologists. The real threat to the island is not ecological disequilibrium, but the inevitable erosion back into the ocean.

Life is the most powerful force in the universe. It will take anything the Earth does to it. But unless we adopt life as our own social paradigm, we will not fare well. If we base our society on control instead of growth, the first unpredictable shock we witness will cause our collapse. So watch Home, be inspired by it, but do yourself a favor and turn the sound off.

This is part I of a series of excerpts of an article to be published in the International Journal of Architectural Research entitled The Principles of Emergent Urbanism. Additional parts will be posted on this blog with the editor’s permission until the complete article appears exclusively in the journal’s upcoming issue.

Of the different domains of design urban design is an oddity. While the design of a machine can be traced to a definite, deliberate act of invention, and even the design of buildings (architecture) is rooted in known production processes, the design of cities was never seriously attempted until well after cities had become a normal, ordinary aspect of civilized living, and while the design of machines and buildings was a conscious effort to solve a particular problem or set of problems, cities appeared in the landscape spontaneously and without conscious effort. This places the efficacy of urban design in doubt. The designers of machines and buildings know fully how the processes that realize their design operate, and this knowledge allows them to predictably conceive the form they are designing. Urban designers do not enjoy such a certainty.

How is it possible for what is obviously a human artifact to arise as if by an act of nature? The theory of a spontaneous order provides an explanation. According to Friedrich A. von Hayek (Hayek, 1973) a spontaneous order arises when multiple actors spontaneously adopt a set of actions that provides them with a competitive advantage, and this behavior creates a pattern that is self-sustaining, attracting more actors and growing the pattern. This takes place without any of the actors being conscious of the creation of this pattern at an individual level. The spontaneous order is a by-product of individuals acting in pursuit of some other end.

In this way cities appear as agglomerations of individually initiated buildings along natural paths of movement, which originally do not require any act of production as dirt paths suffice. As the construction of individual buildings continues the most intensely used natural paths of movement acquire an importance that makes them unbuildable and these paths eventually form the familiar “organic” pattern of streets seen in medieval cities. This process still takes place today in areas where government is weak or dysfunctional, notably in Africa where urban planning often consists of catching up to spontaneous settlement, and in the infamous squatter slums that have proliferated in the 20th century.

A transect of the city of Tultepec in Mexico provides a snapshot of the different phases of spontaneous urban growth. (Google Earth image)

As urbanization becomes denser, the increasing proximity of concurrent, competing individual interests causes conflicts between the inhabitants of the emerging town. Individuals build out their properties in such a way that it interferes with others, for example by blocking paths or views. These acts threaten the sustainability of the spontaneous order, and to resolve this situation the parties involved appeal to the same judges that rule on matters of justice. These judges, again according to Hayek, are required to restore and preserve the spontaneous order with their rulings. These rulings provide the first building regulations and, when government authority becomes powerful enough to do so, are compiled into comprehensive building codes to be applied wherever the force of that government extends. (Hakim, 2001)

The compiled building codes are later brought by colonists to create new settlements, reproducing the morphology across multiple towns but each time in a pattern that is adapted to the local context. Early town planning efforts are attempts at regularizing the building codes in order to plan for long-term organization of cities, but maintain the spontaneous production process. Most notably the rapid urbanization of New York City was accomplished by very simple rules on the size of blocks laid out in the 1811 Commissioners Plan for New York. Unlike the experience of urbanization in previous centuries, where urban growth was slow and often stagnant, the urbanization of New York took place in a time of rapid social and economic changes, and the city government had to invent building codes involving issues that never could arise in a pre-capitalist society: first the tenement, then the skyscraper, and ultimately, the automobile.

Modernism: the replacement for the spontaneous order

Architects and urban planners of the early 20th century, confident in the techniques of engineering and industrial production, believed that the spontaneous city had become irrational and had to be replaced with a new design fully integrating new industrial technology. The Swiss architect Le Corbusier is famous for designing a complete city around the automobile and building models of his design. In so doing he adopted a process of urbanization that was completely planned hierarchically, applying the processes familiar to architects at the scale of an entire city. He also ridiculed the morphology of spontaneous cities as being the product of donkey-paths.

This scale model of Le Corbusier’s Plan Voisin marks the turning point where city plans as constraints on individual initiative are replaced with architectural design at the scale of millions of inhabitants. (Le Corbusier, 1964)

Although the architectural program of high-rise living of Le Corbusier was discovered to be a colossal failure, the modernist process of development replaced spontaneous urbanization in the industrialized world. The housing subdivision substituted adequately for the high-rise tower block, providing affordable housing in large numbers to a war-impoverished society. This production process is still in force today, separating cities into three distinct zones: residential subdivisions, industrial and office parks, and commercial strips.

Modern city planning has been successful at its stated objective, producing a city designed specifically around automobile use, yet it was immediately and has been perpetually the target of criticisms. Most significantly the vocabulary of these criticisms had to be invented in order to spell out the critics’ thoughts because the type of deficiency they were observing had never been seen. Words like placeless or cookie-cutter were invoked but fell on the deaf ears of urban planners who were trained in Cartesian processes and industrial production techniques.

The most devastating criticism of modernist urban planning came in the form of a sociological study and personal defense of the spontaneous city, the book Death and Life of Great American Cities by Jane Jacobs. (Jacobs, 1961) In it she described in great details how the functions of a spontaneous city related and supported each other. Her concluding chapter, the kind of problem a city is, is still the most relevant. In it she attacks the scientific foundations of urban planning at a paradigmatic level, and claims that the methodology of the life sciences, at the time undergoing the revolution created by the discovery of DNA, is the correct approach to studying cities.

Death and Life of Great American Cities has been adopted by contemporary urban planners as a textbook for urbanity. Its descriptions of the characteristics of a city are now the models upon which new developments are planned. The old urban development of housing subdivisions and office parks is being substituted for the new urban development that has streets, blocks, and mixed uses, just as Jacobs had described to be characteristic of life in the city. A major difference between Jane Jacobs’ preferred city and the new urban plans remains. The layout of mixed uses is organized and planned in the same process as Le Corbusier planned his city designs. The scientific suggestions of Jacobs have been ignored.

The discovery of emergence and complexity science

In the time since Jacobs published her attack on planning science molecular biology has made great technological achievements and provided countless insights into the morphology of life. In parallel the computer revolution has transformed the technology of every human activity, including that of design. But the computer revolution brought along some paradigm-altering discoveries along with its powerful technology. In geometry, the sudden abundance of computing power made it possible for Benoit Mandelbrot to investigate recursive functions and his discovery, fractal geometry, generated a universe of patterns that occurred in many aspects of the physical universe as well as living organisms. (Mandelbrot, 1986)

Some thinkers saw that the life sciences were part of a much more general scientific domain. They formed the Santa Fe Institute and under the label complexity studied not only organisms but also groups of organisms, weather systems, abstract computational systems and social systems. This research formed a body of theory called complexity science that has resulted in the creation of similar research institutes in many other places, including some centers dedicated specifically to urban complexity.

Their scientific revolution culminated in two major treatises within the last decade, both from physicists practicing in a field of complexity. The first was A New Kind of Science by computer scientist and mathematician Stephen Wolfram (Wolfram, 2002), where he presents an alternative scientific method necessary to explore the type of processes that traditional science has failed to explain, presenting a theory of the universe as a computational rule system instead of a mathematical system. The second was The Nature of Order (Alexander, 2004) by architect Christopher Alexander, where he presents a theory of morphogenesis for both natural physical phenomena and human productions.

A definition of emergence

To define what is meant by emergence we will use the abstract computational system upon which Wolfram bases his theories, the cellular automaton. Each cell in a row is an actor, making a decision on its next action based on its state and the states of its direct neighbors (its context). All cells share the same rule set to determine how to do this, that is to say all cells will act the same way with the same context. In this way each row is the product of the actions of the cells in a previous row, forming a feedback loop. The patterns of these rows are not in themselves interesting, but when collected in a sequence and displayed as a two-dimensional matrix, they develop complex structures in this dimension.

The 30th rule of all possible rules of one-dimensional cellular automata produces a chaotic fractal when displayed as a two-dimensional matrix, but most other rules do not create complex two-dimensional structures. The first line of the matrix is a single cell that multiplies into three cells in the second line in accordance with the transformation rules pictured below the matrix. This process is reiterated for the change from the second to the third line, and so on. All the information necessary to create structures of this complexity is contained within the rules and the matrix-generating process. (Wolfram, 2002)

The same general principle underlies all other emergent processes. In a biological organism a single cell multiplies into exponentially greater number of cells that share the same DNA rules. These cells create structures in a higher dimension, tissues and organs, which form the entire organism. In the insect world complex nests such as termite colonies emerge from the instinctual behavior of individual termites. And in urbanization, buildings form into shopping streets, industrial quarters and residential neighborhoods, themselves overlapping into a single whole system, the city.

When the modernists unleashed their program for simplifying cities, they did not limit themselves to redirecting existing institutions. Within the modernist ideology was implied the idea that transportation, open space and buildings were separate, isolated things and could therefore be made in isolation. This lead them to create entirely new institutions that operated independently. One of these was the Department of Transportation, also called Ministry of Transportation, or Délégation Départementale de l’Équipement, or somesuch. Its mission was clear: build roads, make them fast, clear away all obstacles and let’s drive. While the collapse of the modernist program in architecture swept away the building typologies they had invented and would later make room for the creation of such architectural artifacts as Poundbury, the transportation system they had devised continues to operate unchallenged. This system, it must be realized, is sprawl. It is the engine of urbanisation throughout the world. It does this by taking rural land and upgrading it, without any conscious realization of what it is doing, into urban land. The very existence of Poundbury, or any other New Urbanist development, is owed to this urbanization. Without it, it could not have any economy. It is therefore incorrect to refer to it as only a transportation system. It is a land production system, creating markets for land where there were none before.

Writer Alex Marshall (not Stephen) correctly diagnosed what was occurring in How Cities Work. In it, he identified the architects and planners of economic powerhouse Silicon Valley as being the state and federal transportation agencies, and invited New Urbanists to work for these agencies if they wanted to have an effective impact on the design of cities. While his book is an illuminating exploration of the processes for building cities and even lays the groundwork for an emergent theory of urbanism, by the middle chapters Alex Marshall seems to lose his mind and lays the blame for the urban mess on market ideology, Republicans and the almighty Libertarians. If only government authority over land was unrestricted, and the voters (I presume) realized that an urban planning system is a choice and not an inevitability, then once again real cities could be made instead of sprawl.

I appreciate his arguments in favor of making better, more informed, richer choices in how cities are planned. But the fact that the tone of his book radically shifts from cool, rational analysis about systems for urbanisation to anger and blame over restrictions imposed on government authority shows that he has disentangled one layer of confusion about cities only to expose another, even more entangled layer: that of land ownership and market creation. To see through that layer requires traveling back in history to the very beginning of the industrial age and farther.

Our ancestors in America lived almost exclusively as homesteaders and independent land owners. The appeal of migrating to the new world was the abundance of homesteadable land. At the beginning of the 19th century, places such as New York City were shipping towns of little importance relative to the agrarian economy. This was not the case in Europe, where the land was owned in very large estates by a small number of aristocratic families. The rest of the population lived upon the land as tenants. Cities also were estates, with those not owned by the aristocracy being owned either by the clergy or by corporations of merchants chartered in the middle ages. (The City of London somehow miraculously continues to operate by this system.)

The fact that Europeans and Americans had a vastly different starting point in land ownership affects the way they conceive of municipal authority. Because land estates always regulated tenants and built improvements such as roads to collect rents, Europeans will see the city as a public service and consider zoning and regulation as completely normal and legitimate. In America every man was his own land owner. Land rents, roads and zoning were imposed through the force of government upon those small land owners. Because of this, Americans see the city as a government interference and resist zoning and regulation.

The suburban dream, to own one’s house, has come to replace the American dream, to own one’s land. But the economic reality of owning a parcel of land and being a landowner are completely different. Our ancestors lived from their land, not simply on it. Their wealth came from their land. A house owner simply resides on his property, he does not live from it. His living is made from his participation in the life of the city, and he needs the roads in order to do that. Should the road to his house be blocked off, either intentionally or by natural accident, the house would become useless but he would not lose his living. A house owner is as much a tenant as his European ancestor. He rents the streets he lives from.

The relationship between transportation and land rents has been known forever. In the late 19th century railroad developers created new neighborhoods of cities by buying up isolated land, extending the railroad to it and then parcellating it into lots. The people of this new neighborhood lived from the railroad commute. The development of Los Angeles is famous for having been driven by streetcar companies expanding to new areas and profiting from the land they improved. The ownership of the streets was then transferred back to city corporations. It was moved from private to communal ownership, with all the pitfalls this implied.

Seeking to increase the value of their developments by adding a more restrictive layer of controls over the subdivisions they produce, developers have for the last few decades been creating home owners’ associations to provide for the roads and control covenants upon the private parcels of home owners. In doing so they have chosen to take an alternative course to municipal incorporation, but unfortunately the organizations have shown themselves to be draconian in their application of rules as banal as the height of lawns and in so doing compounded the reputation of suburban subdivisions as zones of conformity. The HOA is a cooperative ownership model that transforms subdivisions into structures as rigid as a condominium tower, without the material rigidity of the tower itself. If one is an advocate of the suburban subdivision in the name of freedom, this model seems to miss the mark.

It is indisputable that HOAs provide a good, environmental control, that home buyers wish to have. They are willing to give up control over their house in exchange for part control over the neighborhood. The developers are creating neighborhood value by tying their product into a system of rules, and in doing so increasing the demand for the individual homes. Developers are creating rules-constrained markets within the framework of the HOA, a cooperative of home owners who pay a rent to have the right to be part of this market. The HOA market itself exists within the market created by the road that integrates it to the productive economy, road which has almost certainly been created by a municipality or county administration. It is a market within a market, and both have their specific rules as to how land may be utilized.

Alex Marshall claims that market creation is the exclusive domain of government. I believe this is dangerously incorrect, not out of anti-government bias but because the organization of government places strict limits on the complexity of the markets that will be created, something that we both appear to agree is a major flaw of our urbanism. To show this, I will explain the history of one of the weirdest cities of Paris, the business district La Défense.

The most peculiar aspect of the city is its organization of land. It was founded in the late 1950’s and planned in the early 60’s in the model of utopian modernism, which meant strict separation of pedestrian and vehicular traffic and rows of identical tower blocks. The master plan rapidly fell apart and was abandoned when it was realized that the companies for whom the towers were meant had no use for them in their planned shape. From that moment they were free to define what kind of tower they wanted to build, however the system of traffic separation remained. The developer proceeded to build an enormous structure (about three stories tall) over the natural ground called, roughly translated, The Slab. The Slab is the main pedestrian space that integrates all the towers into the mass transit network of metropolitan Paris and the commercial activity of the district. The centre of The Slab (seen left) turns out to be the ceiling of the gigantic mass transit station (seen middle) that distributes 150,000 people to and from their offices morning and night. The buildings at La Défense do not have any connection to land or streets, they float upon The Slab.

This project is said to be one of the flagships of the French state, but the interesting thing about this is that the state had to go around its own system of land administration in order to create it. Municipalities in France are organized around 36,600 communes whose statutes regarding urban planning and land use are strictly defined in the legislation of the administration. La Défense does not have an administration in the same sense that communes do. It was created as a national interest operation, meaning that the state, realizing that its own territorial administration system could not deal with a project of such complexity, declared a zone of territorial exception where the legislation was suspended. Instead, the land inside the zone was purchased by a developer, the EPAD, whose statutes are defined by private business law but whose majority shareholder is the state. In other words, the state had to go through the market to avoid flaws in the system of government in order to create a market unlike anything the country had seen before.

The EPAD has since operated profitably by investing in land improvements, most importantly paying to bring regional rail and metro links to the district, and selling development rights for towers that are limited by contract to certain open-ended morphologies. No one owns land there except EPAD, but private companies do own their buildings. Those buildings never could have been built had EPAD not created a market for them through the expansion of the mass transit network it paid for, and of course by building The Slab.

EPAD, an organization who was meant to last at most 20 years, celebrated its 50th anniversary this year while many people struggled with the “outlaw” nature of its existence. Proposals for reform all crash against the challenge of The Slab and how it could continue to function as a form of urbanism within the framework of standard territorial law. This territorial law is itself being questioned and most likely headed for reform, although there is no guarantee that reformed law will be competent to handle a structure of this nature. The way things are going, it’s possible that EPAD could have a longevity comparable to the City of London.

The lesson of La Défense is that for-profit enterprises can create urbanism, markets for land supported by large capital structures, provided that the law allow them to. In other economic sectors this arrangement is nothing unusual. To persuade Republicans and Libertarians of the benefit of urbanism, it suffices to make a comparison with the New York Stock Exchange company, an icon of capitalism. While the service the company provides is a means to trade stocks efficiently, this service does not come free. The NYSE occupies some valuable real estate in downtown Manhattan, and provides sophisticated computer equipment and trading services to participants. You must pay for your right to take part in the NYSE, and then obviously must pay again to buy shares. Your ability to trade shares may not be possible without the supporting capital investments of the NYSE company. The NYSE, although it creates a market, is itself competing in a bigger market, that of stock exchanges. It must be better at enabling stock trading than other stock exchanges in order to attract traders and companies, and it can itself be bought, sold, merged or spun-off as a company. The stock trades are a market that is nested within the market for stock exchanges.

For the liberal-democrat wary of such fine capitalist institutions, the work of market creation can also be done by a non-profit organization. This is what one of my favorite examples, the Wikimedia Foundation, does with its different wikis. Although the trades are free, the foundation does provide the framework for people to rapidly and easily share and structure information. A foundation which owns land could in the same way create a market for buildings by enabling its inhabitants. It is exactly such a process that Christopher Alexander described in The Oregon Experiment. By his proposal, the University of Oregon was required to provide design assistance to faculty and students who wanted to initiate building projects. This form of market would obviously put enormous power in the hands of the users and less so in those of big developers, which is the opposite of what municipal planning systems have done.

Municipal planning systems, as we have known them for the last two centuries in America, have been held up by two pillars. The grid of streets and later on the supergrid of arterials have urbanized land in a manner that favored land speculators and developers, and then the large land speculators and developers. The second pillar has been the zoning ordinance and building code, a system of bureaucratic regulations that has grown in size to the point where consultants must be hired to navigate the process. The non-professional who wants to build something in this market, if he has been lucky enough to benefit from the grid, must first understand a stack of regulations that will take him enormous effort to master. This cost evidently favors developers who have a lot to invest in building and discourages the creation of the small events, quirky house, shops and businesses, that make a city so pleasurable. Bureaucratic regulations also have the nasty side-effect of being unable to translate qualitative standards into rules, the result being a long list of quantitative regulations that produce big subdivisions, big malls and big business. The SmartCode, 60 pages long, does not improve upon this process. It makes the big subdivisions take a different shape.

The 16th borough of Paris (front) and La Défense (back). Two different markets creating two different emergent morphologies, and different environmental qualities as well. Those are choices.

The creation of the municipal system in Europe was an act of land reform, breaking down land estates that had their origins in feudal privilege. The French communes were the product of the revolution, and the municipal corporations acts of Great Britain in the 19th century put an end to quite a few “rotten boroughs.” The end of the feudal tenure system could be justly seen as progress by Europeans, and the new powers wielded by municipalities did not exceed those wielded by the previous landlords. In America the incorporation of land into municipalities, and the transfer of ever greater powers to county governments, took place at the expense of small land owners who considered themselves independent and resisted the limitations imposed upon them by new municipal regulations, while profiting at the same time from the new markets created for their land by investment in structural improvements, most of all roads. Both processes for creating municipalities have since been the monopoly of governments, and the slow action in creating, dissolving, merging and splitting cities has caused crisis in all areas. Alex Marshall points out that state governments have the authority to create and merge municipal corporations, and should make use of this authority more often, but governments only have limited attention to devote to these issues. In the early 19th century it required an act of government to charter any business corporation. The regularization of this system allowed businesses to incorporate without going through the legislative process, and the dynamic market economy we have enjoyed since has this process as one of its foundations. For some reason it was never thought of to extend the same rights to city corporations.

In their pleas to “leave things to the market” Americans severely restricted what kind of operations municipalities could undertake, and limited their authority to take action against public problems. In doing so they did not realize what kind of market was being created, and who would benefit most from it. A municipality that does more is no less of a market than one that does less. A municipality that builds a hierarchical grid of differently-scaled streets and helps non-professionals to create their own buildings would be creating a market, and ultimately an emergent urban morphology, vastly different from what limited municipalities have been allowed to create. More restrictive rules could also create environmental quality that would raise the value of the market as a whole. The municipality could be creating a better market than it currently is if it had more freedom to act.

Inventing and organizing such a municipality will itself require a process that systems of government cannot handle. This is where things truly ought to be left to the market. Independently owned and operated cities could adapt and reorganize themselves to experiment with new forms of markets. They could merge and spin-off to meet the challenges of metropolitan scale. They could buy the land that they intend to urbanize. They could regulate the land to preserve the quality of the environment and create greater neighborhood value. They could make the immense capital investments in roads, mass transit, public squares, parks and utilities that would earn them a profit in increased rents from their markets. Finally, the disintegrated land creation institutions brought about by the modernist program could be reintegrated in cities with enough authority to employ them efficiently and productively.

Sprawl would soon after be a footnote of history, along with many other urban ills.

Fractal geometry has infiltrated popular culture since it was formalized in the early 80’s from the works of Benoit Mandelbrot. While it has been used to study the form of cities by researchers such as Pierre Frankhauser and Michael Batty, the insights to be drawn from this field of mathematics have not yet penetrated the field of urbanism, defined as the construction of cities. Connecting the fractal city by Nikos Salingaros approaches the topic by asking what type of city is fractal, without going into depth as to how a fractal is made. Christopher Alexander, in his second tome of The Nature of Order, The Process of Creating Life, begins to develop profound ideas on the topic, which he had hinted to in The Oregon Experiment and A New Theory of Urban Design.

The basic quality of fractal geometry is that it is recursively-defined geometry; it must be described in terms of itself. A triangle, in basic euclidean geometry, is defined by the connection of three vectors at their extremities. Euclidean geometry is built up by combining basic elements into different shapes. A point becomes a line, which becomes a triangle, which becomes several different kinds of polygons, and so on. (A famous introductory architecture textbook, Architecture: Form, Space and Order by Francis D. K. Ching uses this method.) Fractal geometry does not take this approach of combination. Instead of using a triangle to make a square, in fractal geometry we use a triangle to make another triangle, such as this Sierpinski triangle:

At each step we use the results of the previous step and repeat some procedure, in this case either adding two copies of the previous object below the current one (composition) or replacing the three large triangles each by a copy of the object (decomposition). Both approaches will generate the Sierpinski triangle over an infinite number of repetitions.

The words generate and infinite are very important. It is these two words that make fractal geometry so completely different from euclidean geometry, which can be drawn instantaneously. Because fractal geometry is recursive, it is in theory infinitely complex, and the only way to see what a fractal object will look like is to run the computation that generates it until we grow tired of watching the process unfold. It is, by its own nature, surprising, unpredictable, and thus emergent.

The idea of objects substituting themselves for copies of themselves is nothing that revolutionary. It is the basic process that underlies all living things. In a living system a starting point, the embryo, contains a program, DNA, that will be multiplied into trillions of cells. The cells follow the transformations described by their DNA codes by taking certain actions depending on their environmental factors and previous states. (Alexander uses the example of a bone, whose shape evenly distributes structural stress across its surface, by claiming that the form of a bone emerges from a program telling cells to add bone mass where the stress is most intense.) Because living systems are the result of recursive transformations, it should not be a shock that they exhibit the properties of fractal geometry. The inward-out, decentralized growth of living things makes possible complexity in nature. Benoit Mandelbrot made this obvious when he wrote The Fractal Geometry of Nature, a book that pretty much started the fractal revolution by providing a mathematical framework for understanding real physical space.

Coming back to our preferred subject matter, cities and their construction, there is something very profound going on in the construction of cities if Mr. Frankhauser and Batty can calculate an index of “fractality” that is higher for some cities than for others. It means that in the process of building cities humans have unconsciously created complexity by adopting certain processes at certain times, and forgotten or abolished them at other times. That is a fascinating topic of discussion, a debate which has been at the heart of the profession going back perhaps further than the vastly different paths taken by Haussmann and Cerda in the 19th century, and one that must be at the heart of the profession of urbanism today more than ever. What is the alternative to planning and deliberate design of cities, which have nothing but a history of failure to show for themselves?

I’ve explained in a previous article how the very purpose of building cities is to create networks of buildings that handle chaos, the everyday uncertainty of future needs against the permanence of individual buildings. The very act of growing a city is an act of differentiation, creating something different from what currently exists as part of the city’s network of buildings in order to fulfill a need that the existing building stock cannot fulfill. (In other words, adaptation to changing circumstances.) These differentiations can be the creation of new public spaces, such as the boulevard construction initiated by Haussmann that provided large-scale connectivity to the city of Paris, as well as the construction of new, unforeseen buildings. But this is where the architectural design approach to urbanism runs into a major problem: how can beauty and order be created out of something that must necessarily be different from everything else?

The answer to that question is hidden in Benoit Mandelbrot’s greatest discovery, the Mandelbrot Set.

The algorithm that generates the Mandelbrot Set is, like those behind all the beautiful complex structures, extremely simple. It is a chaotic algorithm that “spins” within the boundaries of 2 and -2. For given coordinates in the plane made up of the normal and complex numbers, each coordinate will either spin forever in the orbit of radius 2, or escape after a determined number of iterations. The coordinates which never escape are defined as being part of the set.

A black-on-white picture of the set is by itself very intriguing, but the true beauty of it is not revealed until we apply a system of transformation to the coordinates that were thrown out of it. If, each time we throw out a pair of coordinates, we assign to it a number equivalent to the number of iterations it took to figure out it didn’t belong in the set, we will form groups of chaotic equivalence. And once we apply a single, shared transformation (a “DNA code” for the chaotic equation) to these sets, in this case defining a color for each iteration that threw out some coordinates, applying this color to these coordinates while drawing the Mandelbrot Set, we will generate this kind of geometry:

This is what I refer to as structured chaos. By applying a shared system of transformation to chaotic events, we obtain complex geometry. Shared transformations are the source of the new symmetric property of fractals known as self-similarity, and they are also the source of the wholeness and beauty of those chaotic systems called life, including cities.

Reflecting on the way cities have been built throughout history, the most beautiful places have been those that have shared transformations while creating differentiations. The city of Venice, which continues to inspire architects despite their inability to live up to its beauty, is a perfect example of shared transformations creating wholeness out of chaos. But to understand how to create symmetry by self-similarity, one has to be able to decompose buildings into their different scales and chaotic fields (differentiated elements).

The tradition of teaching the classical orders in architecture was once an imperfect approach to granting architects this skill. The classical orders are one form of transformation system, where large-scale elements, the column, the entablature, are decomposed into smaller-scale elements, the capital, the shaft, which form the large scale elements. And so when many architects, trained to share this transformation system as part of their skill set, worked on completely different buildings, their work could easily form a larger whole; whenever they hit similar problems, they would employ the similar solution they were trained to employ. While two buildings may have completely different sizes or roofs, or one could have a bell tower while the other didn’t, if both buildings had windows and columns, the windows and columns would be made the same way, and thus symmetrical to each other. This is how every building in a city was tied together in a web of geometric relationships, and it is the density of these relationships that gave cities their quality of wholeness and beauty. This property goes beyond the scale of the classical orders. It is also true of ancient Asian cities, or the mythic New York of the 1940’s.

Sadly the unending race for pure originality and the abandonment of hierarchical geometry by the architectural profession has made the creation of such cityscapes impossible. This has made the modern architectural profession largely parasitic of the city, and their professional ruin easily explained. Given what we now understand about generating geometric wholeness out of chaos, is there anything that justifies, other than a desire for euclidean perfection, the creation of rigid euclidean plans for cities? I have not been able to find any. The work of urbanism must be about two fundamental aspects: defining a system of transformations that will apply to all unforeseeable acts of construction in the city, at all scales, and creating the connective public space that will bind the different buildings together.

Returning to Nikos Salingaros’ question, the kind of a city that is a fractal is the kind that is made by applying shared transformations to chaotic events. This is the holy grail of urbanism in the 21st century. With this knowledge we can finally surpass the classical city, bury the demons of Le Corbusier and the C.I.A.M. while embracing all that technology has to offer to urbanism.

20th century professional urbanism is the story of a war on complexity in order to control urbanization.

The modernists rebelled against the “mess” of the city. They put everything in their place. In this square shall be the houses. In that square the offices. In that square the stores. In some form of another, this system, called zoning, is in force over 99% of the American continent. Its main advantage is that it is incredibly lazy.

They call this urban planning. They took control of the city’s future by destroying it. But they didn’t really know what they were destroying.

A city is not reducible to parts. A city is a mesh of relationships between spaces. It begins once a space is built to provide a specialized function that is not fulfilled by another existing space, and the two spaces are linked together by a communication system. Let’s call this first space a and the new space b. Once a and b form relationship a-b, the city X is born. X is a set which contains relationships.

When a and b are deficient in some manner, a third space, c, is added to the set. X then becomes (a-b, c-b). Then space d may be added to form the set (a-b, c-b, c-d). This process continues as more spaces are created and new relationships are formed. The city becomes a very complex mesh, or semi-lattice. You cannot isolate any part of this mesh from the rest.

Sim City 4 pictures the shape of these relationships when you click on the commutes for any building. Each building has a web extending out through the city, and these webs overlap and interweave each other as a single system.

The relationships do not split up into group. You cannot define “sub-cities,” groups of relationships independent from each other. You cannot say that city A is made up of building set B and C. Inevitably some buildings in either group will need to form relationships to each other. But this is exactly what zoning is meant to prevent! In doing so, zoning destroys many forms of exchange and holds back the complexity of the city.

What exactly are such relationships? Any reason you might have to get out of the house. It could be going to the bakery. Your house d would form a relationship with bakery f, d-f. The bakery would have many customers in the neighborhood, and they would form relationships f-g, f-h, f-i and so on, even though you may never meet any of them. These people will have jobs that will form relationships g-m, h-n, i-n. All of you, together, create the life of the city, though you may never run into each other. Without business m, bakery f may not have enough customers to continue, and then you would no longer have access to a bakery.

Sometimes a space will lose all of its relationships and will be destroyed, but all the other relationships will remain part of the set. The continuous mesh of relationships is itself fully permanent. This is why cities have names that last through millennia, such as London and Paris, even though every building that made them up at their beginning has long since been removed and forgotten. The set of relationships is still exactly where it has always been. It has been transformed and developed, but never destroyed. At every point in time the set exists even though spaces flow in and out of it, much like a river is not defined as a lump of water molecules but the flow of them.

It is only relationships and not the individual spaces that form a city. A block of identical row houses will not form relationships. Relationships will only form between spaces that are complementary, that is to say spaces that are differently adapted to their own specific functions. It thus makes no sense to create zoning codes for identical houses as there is no reason for these houses to be near each other. However, it does make sense to create multiple houses around a playground, as these houses will form a relationship with the playground.

Defining a city as relationships allows us to differentiate cities which are alive and growing from cities which are dead or dying. When the number of relationships in a city is increasing or stable, the city is alive. When the number of relationships in a city is shrinking or zero, the city is dead, despite the fact that there may still be buildings there! A ghost town does not have relationships.

Good urbanism is the creation of support systems for building relationships. Streets, public spaces, transportation networks and building codes achieve this. Zoning kills them.

The best support systems, the best urbanism, will permit the greatest density of relationships (not density of people), implying the greatest spacial complexity and diversity achievable.